Method and apparatus for cut-point frontier selection and for counter-example generation in formal equivalence verification

ABSTRACT

An approach for cut-point frontier selection and/or counter-example generation. Both a lazy and an eager cut-point frontier are identified. A reconvergence (or non-reconvergence) ratio is then computed for each of the frontiers and the one with the smaller (larger) reconvergence (non-reconvergence) ratio is selected as the next cut-point frontier. For another aspect, to generate a counter-example, in response to identifying a difference in output signals for a given cut-point frontier, values of eigenvariables and reconverging primary inputs are used to compute the corresponding values of the non-reconverging primary inputs. These corresponding values are then computed to be compatible with the internal signal values implied by the cut-point frontier selections that were made to expose the difference in the outputs.

CROSS-REFERENCE TO RELATED APPLICATION

The present application is related to U.S. patent application Ser. No. 09/734,380 entitled, “Method and System for Formal Verification of a Circuit Model,” filed Dec. 11, 2000 and assigned to the assignee of the present invention.

FIELD

An embodiment of the present invention relates to the field of integrated circuit formal equivalence verification, and, more particularly, to an approach for determining a cut-point frontier and/or for counter-example generation.

BACKGROUND

Formal verification typically employs the use of mathematical techniques to “formally” (i.e. without simulating circuit input vectors) compare two circuit design models at the same or differing levels of abstraction to verify logic functionality equivalence between the models. For example, a formal equivalence verification process may be used to compare a logic circuit model in some hardware description language (HDL) against its implementation as a corresponding schematic model at the transistor level. The circuit model at the higher level of abstraction will be referred to herein as the specification circuit model (or spec) while the one at the lower level of abstraction will be referred to herein as the implementation circuit model (or imp).

Formal equivalence verification tools are frequently based on classical binary decision diagrams (BDDs). Classical BDDs are directed acyclic graph structures that encode the value of a Boolean logic function for all possible input value combinations. BDDs, in general, simplify the task of determining Boolean function equivalence because efficient algorithms for equivalence checking and other Boolean operations exist for BDDs. Because the size of a BDD representing a given circuit can grow exponentially larger with respect to the number of inputs and gates in the circuit, the memory requirements for using a formal equivalence verification tool that employs a classical BDD approach may be prohibitively large.

To address this issue, solutions using a divide-and-conquer approach have been developed. Such techniques attempt to partition the specification and implementation circuit models along frontiers of equivalent signal pairs called cut-points. The resulting sub-circuit partitions of each circuit model are then independently analyzed such that the verification task is broken into more manageable units. The manner in which cut-points and/or cut-point frontiers are selected can affect the performance of the associated formal equivalence verification tool.

One problem that may arise for some prior formal verification tools when introducing cut-points into circuit models is that the verification method may return a false negative. A false negative in this context is an indication by the formal equivalence verification tool that two circuits are different, when in reality, they are not. The process of eliminating false negatives can be time consuming.

Further, when a difference between two circuit models is identified, such prior tools typically do not provide the integrated circuit designer with information as to the source of the differences between two circuit models.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is illustrated by way of example and not limitation in the figures of the accompanying drawings in which like references indicate similar elements, and in which:

FIG. 1 is a block diagram of an exemplary system in which the cut-point frontier selection and/or counter-example generation method and/or apparatus of some embodiments may be advantageously used.

FIG. 2 is an exemplary representation of two circuit models to be compared using a formal verification approach of one embodiment.

FIG. 3 is a schematic diagram of a first exemplary circuit portion that may be evaluated using the cut-point frontier selection approach and/or counter-example generation approach of one embodiment.

FIG. 4 is a schematic diagram of a second exemplary circuit portion that may be evaluated using the cut-point frontier selection approach and/or counter-example generation approach of one embodiment.

FIG. 5 is a flow diagram illustrating the formal equivalence verification approach of one embodiment.

FIG. 6 is a flow diagram illustrating the cut-point frontier selection approach of one embodiment.

FIG. 7 is an exemplary representation of the combination of the two circuit models of FIG. 2 showing example cut-point candidates and an example cut-point frontier.

FIG. 8 is a diagram illustrating example eager and lazy cut-point frontiers that may be selected according to one embodiment.

FIG. 9 is a schematic diagram of two exemplary circuit portions that may be compared and for which a counter-example may be generated in accordance with one embodiment.

FIG. 10 is a flow diagram illustrating the counter-example generation approach of one embodiment.

DETAILED DESCRIPTION

A method and apparatus for selecting a cut-point frontier and/or for counter-example generation are described. In the following description, particular types of circuit models, circuits, software tools and systems are described for purposes of illustration. It will be appreciated, however, that other embodiments are applicable to other types of integrated circuit models, circuits, software tools and/or systems.

For one embodiment, both a lazy cut-point frontier and an eager cut-point frontier are identified. A lazy cut-point frontier, as the term is used herein, refers to a frontier among identified cut-points that lies closest to the previous frontier (which may be the inputs of the circuit model or the portion of the circuit model being analyzed), while an eager cut-point frontier corresponds to cut-points closest to the outputs of the circuit model or portion of the circuit model being analyzed. A cut-point frontier is a minimal set of nodes that separates a circuit of interest into two parts such that no signal path from inputs to outputs can exist without going through a node in the frontier.

Once the lazy and eager cut-point frontiers are identified, a reconvergence ratio is computed for each of the frontiers and the one with the smaller reconvergence ratio is selected as the next cut-point frontier. Alternatively, a non-reconvergence ratio may be computed for each of the frontiers and the one with the larger non-reconvergence ratio is selected as the next cut-point frontier as described in more detail below.

For another embodiment, to generate a counter-example, in response to identifying a difference in output signals for a given cut-point frontier, values of eigenvariables and reconverging primary inputs are used to compute the corresponding values of the non-reconverging primary inputs. These corresponding values are computed to be compatible with the internal signal values implied by the cut-point assignment that was selected to expose the difference in the outputs. Counter-example, as the term is used herein, refers to an identification of variable values, for example, that cause a difference in circuits and/or circuit models to be revealed.

Additional details of these and other embodiments are provided in the description that follows.

FIG. 1 is a block diagram of a computer system 100 on which the cut-point frontier selection and/or counter-example generation method and apparatus of one or more embodiments may be advantageously implemented. For this embodiment, the computer system 100 may be a personal or workstation computer system running one of a LINUX, UNIX, or Windows™ operating system, for example. (Windows operating systems are provided by Microsoft Corporation of Redmond, Wash.) Other types of computers and/or computer systems and/or computer systems running other types of operating systems are within the scope of various embodiments.

The computer system 100 includes a processor 105 to execute instructions using an execution unit 110. A cache memory 115 may be coupled to or integrated with the processor 105 to store recently and/or frequently used instructions. The processor 105 is coupled to a bus 120 to communicate information between the processor 105 and other components in the computer system 100.

For one embodiment, the processor 105 is a microprocessor. For other embodiments, however, the processor may be a different type of processor such as, for example, a microcontroller, a digital signal processor, etc.

Also coupled to the bus 120 are one or more input devices 125, such as a keyboard and/or a cursor control device, one or more output devices 130, such as a monitor and/or printer, one or more memories 135 (e.g. random access memory (RAM), read only memory (ROM), etc.), other devices 140 (e.g. memory controller, graphics controller, bus bridge, etc.), and one or more mass storage devices and/or network connectivity devices 145.

The mass storage device(s) and/or network connectivity devices 145 may include a hard disk drive, a compact disc read only memory (CD ROM) drive, an optical disk drive and/or a network connector to couple the computer system 100 to one or more other computer systems or mass storage devices over a network, for example. Further, the mass storage device(s) 145 may include additional or alternate mass storage device(s) that are accessible by the computer system 100 over a network (not shown).

A corresponding data storage medium (or media) 150 (also referred to as a computer-accessible storage medium) may be used to store instructions, data and/or one or more programs to be executed by the processor 100. For one embodiment, the computer-accessible storage medium (or media) 150 stores information, instructions and/or programs 155-189 that, when accessed or executed by the processor 100 or another machine, are used to perform cut-point frontier selection and/or counter-example generation and related actions in accordance with one or more embodiments or to provide output data related to such actions.

For the exemplary embodiment shown in FIG. 1, for example, a formal equivalence verification tool 155 receives data 160 related to a specification circuit model and data 165 related to an implementation circuit model. The data 160 related to the specification circuit model may include, for example, the specification of a logic circuit in some hardware description language (HDL) while the data 165 related to an implementation circuit model may include, for example, what is intended to be a schematic model of the same circuit at the transistor level.

In response to receiving such data and, in some cases, user input via an input device 125, the formal equivalence verification tool 155 then provides output data that may include, for example, an indication of whether the two circuit models being compared are equivalent 170, cut-point and/or cut-point frontier information 175 and/or counter-example information 180.

For one embodiment, the formal equivalence verification tool 155 is a formal equivalence verification tool that prevents the generation of false negatives using normalized binary decision diagrams (BDDs) in accordance with copending U.S. patent application Ser. No. 09/734,380 entitled, “Method and System for Formal Verification of a Circuit Model,” filed Dec. 11, 2000 and assigned to the assignee of the present invention. For such an embodiment, the formal equivalence verification tool 155 may include a normalized BDD computation engine 185, a cut-point frontier selection engine 187 and a counter-example generation engine 189. The function and operation of each of the engines 185, 187 and 189 are described in more detail below. It will be appreciated that the formal equivalence verification tool of other embodiments may not include all of the modules shown in FIG. 1 and/or may include additional modules not shown in FIG. 1.

The concept of normalized BDDs is described with reference to FIGS. 1 and 2. For purposes of illustration, to compare the specification model computing the logic function Gs and the implementation model computing the logic function Gi, for one embodiment, the formal equivalence verification tool 155 creates the miter circuit shown in FIG. 2 and checks whether Gm always produces the logic Boolean value False for all possible values at the circuits' primary inputs X. To do this, the formal verification tool 155 computes the classical BDD of Gm. Because the classical BDD for Gm may be prohibitively large, however, its computation is performed in terms of normalized BDDs of cut-points that belong to the selected cut-point frontiers.

For example, as shown in FIG. 2, the classical BDD for Gm is computed in terms of the normalized BDDs of the cut-points on frontier F2. The normalized BDDs of the cut-points on the frontier F2 are computed by normalizing their classical BDDs, which are, for this example, computed in terms of the normalized BDDs of the cut-points on frontier F1. The normalized BDDs of the cut-points on the frontier F1 are similarly computed for this example by normalizing their classical BDDs that are built in terms of the primary inputs X of the modeled circuits. By following this approach, smaller parts of the original circuit models may be examined such that the sizes of the resulting classical and normalized BDDs are kept smaller during the formal equivalence verification checking process.

With continuing reference to FIG. 2, the process of BDD normalization is described in more detail in reference to an exemplary cut-point Hi on the cut-point frontier F2 having a classical BDD H_(i)({right arrow over (v)}) where {right arrow over (v)} are BDD variables from F1. H_(i)({right arrow over (v)}) can be written as H_(i)({right arrow over (n)},{right arrow over (r)}) where {right arrow over (n)} are those variables from {right arrow over (v)} that do not appear in the classic BDD of another cut-point on the frontier and {right arrow over (r)} are the remaining variables. The {right arrow over (n)} and the {right arrow over (r)} variables are called the non-reconvergent and the reconvergent variables, respectively.

To illustrate the concept of reconvergent and non-reconvergent signals, reference is made to the exemplary circuit portions 300 and 400 of FIGS. 3 and 4, respectively, that may, for example, provide the signals Gs and Gi in FIG. 2. It is assumed, for purposes of example, that a selected cut-point frontier for the circuit portions 300 and 400 includes the nodes Y1, Y2, Z1 and Z2. A reconvergent signal or variable with respect to a cut-point is a signal that has a fanout leading to the circuit's output by going outside of the logic cone of the cut-point signal. Conversely, a non-reconvergent signal or variable is a signal that does not have a fanout leading outside of the logic cone of the cut-point signal towards the circuit's output. Thus, for the exemplary cut-points, the signal X2 is a reconvergent signal/variable with respect to cut-points Y1, Y2, Z1 and Z2, while X1 and X3 are non-reconvergent signals/variables. This concept is also illustrated by the broken lines between cut-points in FIG. 2.

Thus, with continuing reference to FIG. 2, the normalized BDD for H_(i)({right arrow over (n)},{right arrow over (r)}) is then given by the formula e_(i).(∃{right arrow over (n)},H_(i)({right arrow over (n)},{right arrow over (r)}))+(∀{right arrow over (n)},H_(i)({right arrow over (n)},{right arrow over (r)})) where e_(i) is a separate free Boolean variable referred to herein as the eigenvariable of H_(i).

A high-level exemplary method for formal equivalence verification that uses normalized BDDs in accordance with one embodiment is provided by the following pseudo code in which a node N_(i) of an implementation circuit model (e.g. the node G_(i) of FIG. 2) is compared against a node N_(s) of a corresponding specification circuit model (e.g. the node G_(s) of FIG. 2).

Determine equivalence of (N_(s), N_(i)) { i 0 frontier[i] fanin (cone(N_(s))) U fanin (cone(N_(i))) while ((bdd(N_(s)) is not built) or (bdd(N_(i)) is not built)) do build_bdds (cone(N_(s)), cone(N_(i)), frontier[i], size_limit) cut-points cp_ident(cone(N_(s)), cone (N_(i))) if (cut-points is empty) then increase size_limit continue else i i + 1 frontier[i] front_sel(cut-points) end if end while if (bdd(N_(s)) == bdd(N_(i))) then declare_equal( ) else declare_unequal create counter-example (N_(s), N_(i)) end if }

The method implemented by the pseudo code above is further described in reference to a corresponding flow diagram shown in FIG. 5. As shown in FIG. 5, for the comparison of the functions implemented by output nodes of selected specification and implementation circuit models (e.g. N_(s) and N_(i) in the pseudo-code above), the cut-point frontier is initially set at the primary inputs of the cones that are driving the two signals (block 505).

Then, repeatedly, the process attempts to identify more cut-points lying ahead of the current cut-point frontier closer to N_(s) and N_(i) in an attempt to eventually build the BDDs for these two nodes so that they can be compared for equivalence. The function build_bdds in the pseudo code above (block 510), which may be implemented, for example, by the normalized BDD engine 185 of FIG. 1, attempts to build BDDs for the signal nodes ahead of the present frontier closer to the outputs. These BDDs are built not by assigning free variables for the cut-points on the present frontier, but by assigning to every cut-point its corresponding normalized function as discussed above and in the above-reference copending application. While the BDDs for the outputs are not yet built (block 515), the formation of BDDs stops and the build_bdd function exits the first time that some BDD exceeds a pre-specified limit (block 520). This pre-specified limit is indicated by the variable size_limit in the pseudo code above and may, for example, be user-specified.

Once the function build_bdd exits, cut-point indentification (cp_ident) begins (block 525). The function cp_ident employs an associative map in which BDDs are used as keys to identify sets of equivalent nodes between the two circuit models. If no cut-points are identified among the nodes whose BDDs are built within the specified size_limit (block 530), then this limit is increased (block 535) and an attempt is made to build more BDDs ahead of the same cut-point frontier (block 510). Eventually, the set of identified cut-points becomes non-empty and a new cut-point frontier F_(i) is selected (block 540) from the identified cut-points using the front_sel function as illustrated in FIG. 7. In FIG. 7, for purposes of simplicity, the miter circuit 700 is used to represent the miter circuit of FIG. 2.

For one embodiment, the front_sel function selects the new cut-point frontier according to one of three strategies: a lazy strategy, an eager strategy or a combined strategy. In accordance with the lazy strategy, a frontier of cut-points is selected such that is closest to the inputs as compared to other cut-point frontiers that might possibly be selected. In contrast, according to the eager strategy, a cut-point frontier that is closest to the output(s) is selected. Exemplary eager and lazy frontiers for an exemplary set of cut-points are illustrated in FIG. 8.

Cut-point frontiers that are selected in accordance with the lazy strategy generally tend to lead to smaller normalized BDDs, while cut-point frontiers that are selected according to the eager strategy generally tend to lead to a faster overall verification process because verification progresses with larger leaps towards the outputs. The relative performance of these strategies, however, may vary significantly depending on the circuit model being verified.

Thus, for some embodiments, a combination of the two strategies is used as illustrated by the flow chart of FIG. 6. The method illustrated by FIG. 6 may be implemented, for example, by the cut-point frontier selection engine 187 of FIG. 1.

For embodiments for which a combination of the two strategies is used, cut-point frontiers are identified in accordance with both the eager and lazy strategies (blocks 605 and 610) and are referred to herein as lazy and eager cut-point frontiers. It will be appreciated that, for other embodiments, the order of blocks 605 and 610 may be reversed.

For one embodiment, to identify the eager frontier at block 605, from the outputs of the specification and implementation circuit models of interest, a depth first traversal is performed stopping either at primary inputs or cut-points identified as described above. To identify the lazy frontier at block 610, a depth first traversal is performed from the present cut-point frontier towards the outputs stopping at newly found cut-points.

For one embodiment, a reconvergence ratio is then computed for each of the lazy and eager frontiers (block 615). The reconvergence ratio for a frontier is the ratio of reconverging variables over the total number of variables in the BDDs for the cut-points. Once this ratio has been computed for each of the lazy and eager frontiers, the frontier with the smallest reconvergence ratio is selected as the next cut-point frontier (block 620).

For an alternative embodiment, a non-reconvergence ratio may instead be computed (block 615). The non-reconvergence ratio for a frontier is the ratio of non-reconverging variables over the total number of variables in the BDDs for the cut-points associated with the frontier. Where this approach is used, the frontier with the largest non-reconvergence ratio is selected as the next cut-point frontier (block 620).

Referring to FIG. 2 for purposes of illustration, where the frontier F1 is the prior frontier and the frontier F2 is one of the identified eager or lazy cut-point frontiers as described above, the reconvergence (or non-reconvergence) ratio associated with F2 may be calculated in the following manner. The BDDs for each of the cut-points associated with the frontier F2 are analyzed to determine which cut-points from F1 affect them. Based on this analysis, the number of BDDs on F2 in which any cut-point from F1 appears can then be determined. If this number is greater than 1, the variable corresponding to the cut-point on F1 is determined to be a reconvergent variable. Otherwise, the variable is a non-reconvergent variable.

Once all BDDs for the cut-points on F2 have been analyzed in this manner, the reconvergence or non-reconvergence ratio associated with the frontier F2 can be calculated in the manner described above. The reconvergence or non-reconvergence ratio for the other identified cut-point frontier (either lazy or eager) is then calculated in a similar manner, compared to the previously calculated ratio as described above and the desired cut-point frontier is then selected.

Using the above-described combined strategy for cut-point frontier selection, even with the overhead of identifying two separate frontiers and selecting between them, it may be possible to verify larger circuits than with prior approaches and/or it may be possible to perform formal equivalence verification for complex test cases more quickly.

Referring back to FIG. 5, after the selection of the new frontier 540, the process of building additional BDDs is continued (block 510). Eventually, if the BDDs for the outputs of the two cones are built (block 515), they are compared (block 545). The BDDs for the two outputs are derived based on the last selected cut-point frontier and the normalized functions of the cut-points on this frontier. If the BDDs are equal (block 550), the nodes are declared as equal (block 555). If the BDDs are different, however, the nodes are declared as being inequivalent (block 560) and, for one embodiment, a counter-example is generated (block 565) to indicate to the designer, the source of the inequivalence. In this manner, debugging of a faulty circuit may proceed more efficiently.

It will be appreciated that, for embodiments for which the formal verification process inherently avoids the generation of false negatives, a false negative elimination process does not need to be performed in response to identifying nodes as being inequivalent. For such embodiments, any identified inequivalence is due to the presence of a bug. In contrast to other approaches that require the re-substitution of cut-point variables by their driving functions, where normalized BDDs are used as described above, for one embodiment, an efficient approach that does not require an inordinate amount of memory may be used to generate a corresponding counter-example.

The counter-example (cex) generation approach of one embodiment for exemplary nodes Ns and Ni is represented by the following pseudo-code and may be implemented, for example, by the counter-example generation engine 189 of FIG. 1.

Create cex (Ns, Ni) { (line 1) cex satisfy (bdd(Ns) ⊕ bdd(Ni)) (line 2) while (cex contains eigenvariables) do (line 3) v = select_eigenvar (cex) (line 4) f = substitute (cex, bdd(node_of(v)) (line 5) nf= substitute (cex, nbdd(node_of(v)) (line 6) cex1 satisfy (f ⊙ nf) (line 7) cex merge (v, cex, cex1) (line 8) end while (line 9) return cex (line 10)

The counter-example generation approach of one embodiment is now described in more detail. As mentioned above, a counter-example is generated based on the values of eigenvariables and reconverging primary inputs associated with the sequence of cut-point frontier selections that exposed a difference in the circuit outputs. The approach of one embodiment for counter-example generation is described in reference to exemplary circuit portions 900 and 905 of FIG. 9, which are determined to be inequivalent in the following manner.

A cut-point frontier F1 including the cut-points y₁, y₂, z₁ and z₂ is selected as described above in reference to FIG. 6. The classical BDDs for the cut-points are calculated as follows:

BDD→y₁=x₁x₂x₃

y₂=x₄x₅{overscore (x)}₃

z₁=x₁x₂x₃

z₂=x₄x₅{overscore (x)}₃

So frontier F1 includes cut=points C₁=(y₁, z₁) and C₂=(y₂, z₂) with corresponding BDDs x₁x₂x₃ and x₄x₅x₃, respectively. As can be seen, each of the variables x₁, x₂, x₄ and x₅ appear in only one of the BDDs associated with the cut-points C₁ and C₂ on F1 and are, therefore, non-reconvergent variables. The variable x₃, however, appears in two BDDs and is, therefore, a reconvergent variable.

Then, as described above, the classical BDDs for the cut-points C₁ and C₂ on F1 are normalized.

C ₁({right arrow over (n)},{right arrow over (r)})=x ₁ x ₂ x ₃ with {right arrow over (n)} =<x ₁ ,x ₂> and {right arrow over (r)}=<x₃>

C ₂({right arrow over (n)},{right arrow over (r)})=x ₄ x ₅ {overscore (x)} ₃ with {right arrow over (n)}=<x ₄ , x ₅> and {right arrow over (r)}=<x ₃>

The normalized BDD for C₁ is calculated as follows: $\begin{matrix} {\begin{matrix} {{{ei} \cdot \left( {{\exists\overset{\rightarrow}{n}},{C_{1}\left( {\overset{\rightarrow}{n},\overset{\rightarrow}{r}} \right)}} \right)} +} \\ \left( {{\forall\overset{\rightarrow}{n}},{C_{1}\left( {\overset{\rightarrow}{n},\overset{\rightarrow}{r}} \right)}} \right) \end{matrix} = {{{ei} \cdot \left( {{\exists{x_{1}x_{2}}},{x_{1}x_{2}x_{3}}} \right)} + \left( {{\forall{x1x2}},{x_{1}x_{2}x_{3}}} \right)}} \\ {= {{{ei} \cdot \left( {{1 \cdot 1 \cdot x_{3}} + {1 \cdot 0 \cdot x_{3}} + {0 \cdot 1 \cdot x_{3}} + {0 \cdot 0 \cdot x_{3}}} \right)} +}} \\ {\left( {\left( {1 \cdot 1 \cdot x_{3}} \right) \cdot \left( {1 \cdot 0 \cdot x_{3}} \right) \cdot \left( {0 \cdot 1 \cdot x_{3}} \right) \cdot \left( {0 \cdot 0 \cdot x_{3}} \right)} \right)} \\ {= {{e1} \cdot x_{3}}} \end{matrix}$

The normalized BDD for C₂ is calculated in a similar manner and determined to be equal to e2 • {overscore (x)}₃ as eigenvariable e₂ takes the place of e₁, x₄ corresponds to x₁, x₅ corresponds to x₂ and finally {overscore (x)}₃ takes the place of x₃.

So the cut-points C₁ and C₂ on F1 have respectively normalized BDDs e₁ • x₃ and e₂ • {overscore (x)}₃. Based on this, the BDD for

Ws=e ₁ • x ₃ +e ₂ • {overscore (x)} ₃ while the BDD for

Wi=(e ₁ • x ₃)•(e ₂ • {overscore (x)} ₃)=0.

Clearly, Ws≠Wi and, therefore, the associated circuit models are not equivalent. Because the formal equivalence verification of the circuit models was performed in a manner that prevents generation of false-negatives as described above, no additional false negative elimination process is necessary.

With continuing reference to FIG. 9 and the flow chart of FIG. 10, to generate the corresponding counter-example, it is first determined what values of primary input variables variables and cut-point-associated eigenvariables cause an exclusive OR (XOR) of the BDDs for the outputs Ws and Wi of the two models to be true (block 1005) (i.e. what values cause the BDDs of the outputs Ws and Wi of the two models to be different) (line 2 of the pseudo code).

For the circuit model of FIG. 9, Ns in the pseudo code above is Ws with a BDD of e₁ x₃+e₂{overscore (x)}₃ while Ni is Wi with a BDD of 0. So

cex←satisfy ((e ₁ x ₃ +e ₂ {overscore (x)} ₃)⊕ 0)

cex←satisfy (e ₁ x ₃ +e ₂{overscore (x)}₃)

which can happen in one of 2 ways:

either (e ₁=1, x ₃=1)→cex_(—)1, or

(e ₂=1, x ₃=0)→cex_(—)2.

For purposes of example, it is assumed that cex2=<e₂=1, x₃=0> is selected.

At line 3 of the pseudo code, while cex contains an eigenvariable (block 1010), an eigenvariable from cex is selected (block 1015 and line 4). For this example, there is only one eigenvariable e2, which is the eigenvariable for the cut-point C₂ (y₂ or z₂). The variables in the BDD of the node corresponding to the selected eigenvariable are then substituted with their corresponding values from the cex constructed so far. This happens for exactly those variables of the BDD that have been assigned values in the counter example (block 1020 and line 5). So in this example,

node_of (v)=node_of (e ₂)=z ₂ or y ₂

bdd (z ₂)=bdd(y ₂)=x ₄ x ₅ {overscore (x)} ₃

f=substitute (<e ₂=1, x ₃=0> into x ₄ x ₅ {overscore (x)} ₃)=x ₄ x ₅•{overscore (0)}=x ₄ x ₅•1

f=x₄x₅

At line 6 of the pseudo code and block 1025 of FIG. 10, this process is repeated for the normalized BDD. For this example,

node_of(v)=node_of (e ₂)=y ₂ or z ₂ as previously discussed

nbdd(y ₂)=nbdd(z ₂)=e ₂ {overscore (x)} ₃

nf=substitute (<e ₂=1, x ₃=0> into e ₂ {overscore (x)} ₃)

nf=1 •{overscore (0)}=1•1=1

At line 7 of the pseudo code (block 1030 of FIG. 10), the values that cause f and nf to be equal are determined:

cex1←satisfy (f ⊙ nf)

here f=x₄x₅ and nf=1

so f⊙nf=x₄x₅ ⊙ 1=x₄x₅

satisfy (x ₄ x ₅)→x ₄=1, x ₅=1

thus cex1=<x₄=1, x₅=1>

At line 8 of the pseudo code (block 1035 of FIG. 10) the value assignments from cex and cex1 are returned excluding the assignment for the variable v. The goal of this part of the process is to continue to make all the eigenvariable assignments disappear such that, at the end of the process, only assignments at the primary inputs remain. Thus, for this example,

cex←merge (v, cex, cex1)

cex←merge (e ₂ , <e ₂=1, x ₃=0>, <x ₄=1, x ₅=1>)

cex←<x₃=0, x₄=1, x₅=1>

At this point, it is again determined whether cex contains any more eigenvariables (line 3, block 1010). In this example, cex=<x₃=0, x₄=1, x₅=1> and thus, contains no eigenvariables. The counter example is then returned at block 1040 (line 10 of the pseudo code).

Therefore, for this example, the counter example is <x₃=0, x₄=1, x₅=1>, which, regardless of the values for x₁ and x₂ causes Ws=1 and Wi=0.

Using the above-described counter-example generation approach of one embodiment, once two circuit models are determined to be inequivalent, information can be provided back to a circuit designer, for example, to indicate one or more input variable combinations that cause the difference in circuit models to be revealed.

In the foregoing specification, the invention has been described with reference to specific exemplary embodiments thereof. It will, however, be appreciated that various modifications and changes may be made thereto without departing from the broader spirit and scope of the invention as set forth in the appended claims. The specification and drawings are, accordingly, to be regarded in an illustrative rather than a restrictive sense. 

What is claimed is:
 1. A method comprising: for an integrated circuit model, identifying a lazy cut-point frontier and an eager cut-point frontier, the lazy cut-point frontier being one of inputs of the model and a cut-point frontier closest to a prior cut-point frontier and the eager cut-point frontier being a cut-point frontier closest to outputs of the model, each of the lazy and eager cut-point frontiers being associated with a set of variables indicating signal lines that pass through the respective cut-point frontier; computing a ratio of one of reconverging and non-reconverging variables to a total number of variables for each of the identified lazy and eager cut-point frontiers, a reconverging variable identifying a signal line that has a fanout leading to an output of the integrated circuit model by going outside of a logic cone of the signal line, a non-reconverging variable identifying a signal line that does not have a fanout leading to an output of the integrated circuit model by going outside of a logic cone of the signal line; and selecting from the lazy and eager cut-paint frontiers based on the computed ratios.
 2. The method of claim 1 wherein computing a ratio comprises computing a ratio of reconverging variables to a total number of variables (reconvergence ratio), and wherein selecting comprises selecting the cut-point frontier with the smallest reconvergence ratio.
 3. The method of claim 1 wherein computing a ratio comprises computing a ratio of non-reconverging variables to a total number of variables (non-reconvergence ratio), and wherein selecting comprises selecting the cut-point frontier with the largest non-reconvergence ratio.
 4. The method of claim 1 wherein identifying a lazy cut-point frontier and an eager cut-point frontier includes identifying the lazy and eager cut-point frontiers based on cut-points identified in a specification circuit model and an implementation circuit model.
 5. The method of claim 4 wherein identifying a lazy cut-point frontier and an eager cut-point frontier includes identifying the lazy and eager cut-point frontiers based on cut-points identified in a specification circuit model in a hardware description language and a schematic.
 6. A machine-accessible storage medium storing instructions that, when accessed by a machine, cause the machine to perform: identifying a lazy cut-point frontier and an eager cut-point frontier for an integrated circuit model, the lazy cut-point frontier being one of inputs of the model and a cut-point frontier closest to a prior cut-point frontier and the eager cut-point frontier being a cut-point frontier closest to outputs of the model, each of the lazy and eager cut-point frontiers being associated with a set of variables indicating signal lines that pass through the respective cut-point frontier; computing a ratio of one of reconverging and non-reconverging variables to a total number of variables for each of the identified lazy and eager cut-point frontiers, a reconverging variable identifying a signal line that has a fanout leading to an output of the integrated circuit model by going outside of a logic cone of the signal line, a non-reconverging variable identifying a signal line that does not have a fanout leading to an output of the integrated circuit model by going outside of a logic cone of the signal line; and selecting from the lazy and eager cut-point frontiers based on the computed ratios.
 7. The machine-accessible storage medium of claim 6 wherein computing a ratio comprises computing a ratio of reconverging variables to a total number of variables (reconvergence ratio), and wherein selecting comprises selecting the cut-point frontier with the smallest reconvergence ratio.
 8. The machine-accessible storage medium of claim 6 wherein computing a ratio comprises computing a ratio of non-reconverging variables to a total number of variables (non-reconvergence ratio), and wherein selecting comprises selecting the cut-point frontier with the largest non-reconvergence ratio.
 9. An apparatus comprising: an input to receive integrated circuit models to be formally verified; and a cut-point frontier selection engine to identify an eager and a lazy cut-point frontiers, the lazy cut-point frontier being one of inputs of the model and a cut-point frontier closest to a prior cut-point frontier and the eager cut-point frontier being a cut-point frontier closest to outputs of the model, each of the lazy and eager cut-point frontiers being associated with a set of variables indicating signal lines that pass through the respective cut-point frontier, to compute one of a reconvergence and a non-reconvergence ratio for each of the identified cut-point frontiers, a reconvergence ratio being a ratio of reconverging variables to a total number of variables associated with the respective cut-point frontier, a non-reconvergence ration being a ratio of non-reconverging variables to a total number of variables associated with the respective cut-point frontier, a reconverging variable identifying a signal line that has a fanout leading to an output of the integrated circuit model by going outside of a logic cone of the signal line, a non-reconverging variable identifying a signal line that does not have a fanout leading to an output of the integrated circuit model by going outside of a logic cone of the signal line and to select between the identified eager and the lazy cut-point frontiers based on the computed ratios.
 10. The apparatus of claim 9 wherein the cut-point frontier selection engine is to compute a reconvergence ratio for each of the identified cut-paint frontiers and to select the cut-point frontier with the smallest reconvergence ratio.
 11. The apparatus of claim 9 wherein the cut-point frontier selection engine is to compute a non-reconvergence ratio for each of the identified cut-point frontiers end to select the cut-point frontier with the largest non-reconvergence ratio.
 12. The apparatus of claim 9 further comprising: a counter-example generation engine, the counter-example generation engine to identify values of variables for inputs to the circuit models that cause the circuit models to produce results that are different from each other.
 13. The apparatus of claim 12 wherein the counter-example generation engine is to identify the values based on the values of eigenvariables and reconverging primary inputs associated with a sequence of cut-point frontier selections that causes a difference in the outputs of the circuit models.
 14. The apparatus of claim 9 wherein a first one of the integrated circuit models is a specification circuit model and the second one of the circuit models is an implementation circuit model.
 15. The apparatus of claim 14 wherein the specification model is specified in a hardware description language and the implementation circuit model is a schematic. 